The commutativity degree of groups and rings has been studied by certain authors since 1973, and the main result obtained is [Formula: see text], where Pr (A) is the commutativity degree of a non-abelian group (or ring) A. Verifying this inequality for an arbitrary semigroup A is a natural question, and in this paper, by presenting an infinite class of finite non-commutative semigroups, we prove that the commutativity degree may be arbitrarily close to 1. We name this class of semigroups the almost commutative or approximately abelian semigroups.

The non-commuting graph associated to a non-abelian group [Formula: see text], [Formula: see text], is a graph with vertex set [Formula: see text] where distinct non-central elements [Formula: see text] and [Formula: see text] of [Formula: see text] are joined by an edge if and only if [Formula: see text]. The non-commuting graph of a non-abelian finite group has received some attention in existing literature. Recently, many authors have studied the non-commuting graph associated to a non-abelian group. In particular, the authors put forward the following conjectures: Conjecture 1. Let [Formula: see text] and [Formula: see text] be two non-abelian finite groups such that [Formula: see text]. Then [Formula: see text]. Conjecture 2 (AAM’s Conjecture). Let [Formula: see text] be a finite non-abelian simple group and [Formula: see text] be a group such that [Formula: see text]. Then [Formula: see text]. Some authors have proved the first conjecture for some classes of groups (specially for all finite simple groups and non-abelian nilpotent groups with irregular isomorphic non-commuting graphs) but in [Moghaddamfar, About noncommuting graphs, Sib. Math. J. 47(5) (2006) 911–914], Moghaddamfar has shown that it is not true in general with some counterexamples to this conjecture. On the other hand, Solomon and Woldar proved the second conjecture, in [R. Solomon and A. Woldar, Simple groups are characterized by their non-commuting graph, J. Group Theory 16 (2013) 793–824]. In this paper, we will define the same concept for a finite non-commutative Moufang loop [Formula: see text] and try to characterize some finite non-commutative Moufang loops with their non-commuting graph. Particularly, we obtain examples of finite non-associative Moufang loops and finite associative Moufang loops (groups) of the same order which have isomorphic non-commuting graphs. Also, we will obtain some results related to the non-commuting graph of a finite non-commutative Moufang loop. Finally, we give a conjecture stating that the above result is true for all finite simple Moufang loops.

No abstract

For any non-abelian group G, the non-commuting graph of G, Γ = Γ G , is graph with vertex set G\Z(G), where Z(G) is the set of elements of G that commute with every element of G and distinct non-central elements x and y of G are joined by an edge if and only if xy = yx. The non-commuting graph of a finite Moufang loop has been defined by Ahmadidelir. In this paper, we show that the multiple complete split-like graphs and the non-commuting graph of Chein loops of the form M (D 2n , 2) are perfect (but not chordal). Then, we show that the non-commuting graph of a non-abelian group G is split if and only if the non-commuting graph of the Moufang loop M (G, 2) is 3−split. Precisely, we show that the non-commuting graph of the Moufang loop M (G, 2), is 3−split if and only if G is isomorphic to a Frobenius group of order 2n, n is odd, whose Frobenius kernel is abelian of order n. Finally, we calculate the energy of generalized and multiple splite-like graphs, and discuss about the energy and also the number of spanning trees in the case of the non-commuting graph of Chein loops of the form M (D 2n , 2).

No abstract